\[ \left (-4 a^2 \nu ^2+4 a^2 x^{2 a}+1\right ) y'(x)+x^2 y^{(3)}(x)+3 x y''(x)=4 a^3 x^{2 a-1} y(x) \] ✓ Mathematica : cpu = 0.0315956 (sec), leaf count = 102
DSolve[(1 - 4*a^2*nu^2 + 4*a^2*x^(2*a))*Derivative[1][y][x] + 3*x*Derivative[2][y][x] + x^2*Derivative[3][y][x] == 4*a^3*x^(-1 + 2*a)*y[x],y[x],x]
\[\left \{\left \{y(x)\to c_2 \left (x^{2 a}\right )^{-\nu } \, _1F_2\left (-\nu -\frac {1}{2};1-2 \nu ,1-\nu ;-x^{2 a}\right )+c_3 \left (x^{2 a}\right )^{\nu } \, _1F_2\left (\nu -\frac {1}{2};\nu +1,2 \nu +1;-x^{2 a}\right )+c_1 \, _1F_2\left (-\frac {1}{2};1-\nu ,\nu +1;-x^{2 a}\right )\right \}\right \}\] ✓ Maple : cpu = 0.076 (sec), leaf count = 88
dsolve(x^2*diff(diff(diff(y(x),x),x),x)+3*x*diff(diff(y(x),x),x)+(4*a^2*x^(2*a)+1-4*nu^2*a^2)*diff(y(x),x) = 4*a^3*x^(2*a-1)*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} \hypergeom \left (\left [-\frac {1}{2}\right ], \left [-\nu +1, \nu +1\right ], -x^{2 a}\right )+c_{2} x^{-2 a \nu } \hypergeom \left (\left [-\frac {1}{2}-\nu \right ], \left [1-2 \nu , -\nu +1\right ], -x^{2 a}\right )+c_{3} x^{2 a \nu } \hypergeom \left (\left [\nu -\frac {1}{2}\right ], \left [\nu +1, 2 \nu +1\right ], -x^{2 a}\right )\]